Suppose F has the property that if a∈A appears as a a first entry in an element of F, then it is the only element of F that it appears in
Then we call F a function
If (a,b)∈F, we write f(a)=b
In a function, second entries can be matched with more than one first entry
If, F⊆A×B is a function with f(a)=biff(a,b)←F we often omit F and only refer to f and say:
f:A→B bdg-infoWe say f maps A into B
Consider the following example
A &= \Set{0,1,2,3,4,5,6,7,8,9}\\
B &= \Set{r,s,t,u,v,w,x,y,z}\\
F &= \Set{(0,r), (1,s), (2,s), (3,t), (4,u), (5,w), (6,x), (7,x), (8,x), (9,x) }
Then the rule for f may be stated explicitly or inferred from F
bdg-warningNoterange(f)=\Setr,s,t,u,v,w,x,y,z⊂B is a proper subset of B
Let us write f−1(b)= pre-image of b under f=\Seta∈A∣b=f(a)
For our example
f−1(z)=∅ f−1(t)=\Set3,f−1(x)=\Set6,7,8,9
Composition of function
Let f:A→B & g:B→C be two function
We define the composition of g and f to be the function g∘f:A→C defined by g∘f(a)=g(f(a))
Composition of More Than 2 Function
We can define the composition of more than 2 functions if f:A→B, g:B→G, h:G→D, then
h∘g∘f:A→D and h∘g∘f(a)=h(g(f(a))) etc
Inverse Functions
Let f:A→B be a 1-1 onto map
Then for ∀b∈B,∃ exactly one a∈Astf(a)=b
We define the inverse function of f, called f−1(b)=a iff f(a)=b and we say f is invertible
Identity Function
If we let idA:A→A be the identity map on A, idA(a)=a,∀a←A and idB:B→B be the identity map on B,idB(b)=b, then for f, 1-1, onto, f−1∘f=idA, f∘f−1=idB